Finite Element Methods for Engineers Exercise 3
Overview
Repetition Task 1 Task 2
2 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e
3 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type
3 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type 3. Calculate the element stiffness matrix for all elements
3 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type 3. Calculate the element stiffness matrix for all elements 4. Assemble the global stiffness matrix
3 of 12
K
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type 3. Calculate the element stiffness matrix for all elements 4. Assemble the global stiffness matrix 5. Prepare the whole equation system
3 of 12
K r
= Ka
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type 3. Calculate the element stiffness matrix for all elements 4. Assemble the global stiffness matrix 5. Prepare the whole equation system
K r
6. Prepare the reduced equation system
3 of 12
= Ka r ˜
˜ a (Delete rows/columns with 0-displacements) = K˜
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Procedure for the tasks in this exercise 1. Determine all relevant information for each element e 2. Derive/Look up the element stiffness matrices for each used element type 3. Calculate the element stiffness matrix for all elements 4. Assemble the global stiffness matrix 5. Prepare the whole equation system
K r
6. Prepare the reduced equation system
= Ka r ˜
7. Solve the reduced equation system r˜ =
3 of 12
˜ a (Delete rows/columns with 0-displacements) = K˜ ˜a K˜
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Element stiffness matrix of a rod
e
K
4 of 12
EA = L
cos(α)2 cos(α)sin(α) − cos(α)2 − cos(α)sin(α)
cos(α)sin(α) sin(α)2 − cos(α)sin(α) − sin(α)2
2
− cos(α)
− cos(α)sin(α) cos(α)2 cos(α)sin(α)
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
− cos(α)sin(α) − sin(α) 2
cos(α)sin(α) sin(α)2
Overview
Repetition Task 1 Task 2
5 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Task 1
Calculate the displacement and force vector. F 2
E = 210 000 N/mm2, A1 = 25 mm2, L
❦ 1
2❦
y 1
6 of 12
x
α 3
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
A2 α L F
= 50 mm2, = 135 ◦, = 400 mm, = 250 N
Task 1 Solution 1. Determine all relevant information for each element e
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Element e
αe
sin αe
❦ 1 ❦ 2
90◦
√ 1
135
◦
2/2
cos αe
−
√ 0
2/2
Le
Ae
E e
√ L
A1
E
2L
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
1
2A
E
Task 1 Solution 1. Determine all relevant information for each element e Element e
αe
sin αe
❦ 1 ❦ 2
90◦
√ 1
5. Prepare the whole equation system F 1,x F 1,y F 2,x F F 3,x F 3,y
−
7 of 12
135
◦
r
2/2
cos αe
−
√ 0
2/2
Le
Ae
E e
√ L
A1
E
2L
1
2A
E
= Ka
0 0 √ 2A1 E 0 = 0 2L 0 0
√ 0
0 0 √ 2 0 − 2 −1√ 0 1 √ − 2 −1 1 + 2 0 −1 1 0 1 −1
0 0 −1 1 1 −1
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
0 0 1 −1 −1 1
0 0 0
a2,y 0 0
Task 1 Solution 1. Determine all relevant information for each element e Element e
αe
sin αe
❦ 1 ❦ 2
90◦
√ 1
5. Prepare the whole equation system F 1,x F 1,y F 2,x F F 3,x F 3,y
−
135
◦
r
−F =
7 of 12
−
√ 0
2/2
Le
Ae
E e
√ L
A1
E
2L
1
2A
E
= Ka
0 0 √ 2A1 E 0 = 0 2L 0 0
6. Prepare the reduced equation system
2/2
cos αe
√ 0
0 0 √ 2 0 − 2 −1√ 0 1 √ − 2 −1 1 + 2 0 −1 1 0 1 −1
√ =
r ˜
0 0 −1 1 1 −1
˜a K˜
0 0 1 −1 −1 1
√ 2A1E (1 + 2)a2,y 2L
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
0 0 0
a2,y 0 0
Task 1 Solution 1. Determine all relevant information for each element e Element e
αe
sin αe
❦ 1 ❦ 2
90◦
√ 1
5. Prepare the whole equation system F 1,x F 1,y F 2,x F F 3,x F 3,y
−
135
◦
r
0 0 √ 2A1 E 0 = 0 2L 0 0
−F =
−
√ 0
2/2
Le
Ae
E e
√ L
A1
E
2L
1
2A
E
= Ka
6. Prepare the reduced equation system
2/2
cos αe
√ 0
0 0 √ 2 0 − 2 −1√ 0 1 √ − 2 −1 1 + 2 0 −1 1 0 1 −1
√ =
r ˜
7. Solve the reduced equation system r˜ =
0 0 −1 1 1 −1
˜a K˜
0 0 1 −1 −1 1
√ 2A1E (1 + 2)a2,y 2L
˜a K˜
a2,y = −1.1158 × 10−2 mm 7 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
0 0 0
a2,y 0 0
Task 1 Solution
For the complete vectors follows:
r
8 of 12
=
−−
0 146.45 103.55 250 103.55 103.55
N,
a
=
−
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
0 0 0 1.1158 0 0
·
10−2 mm
Overview
Repetition Task 1 Task 2
9 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Task 2
Calculate the displacement and force vector. F
❦ 1
1
2
❦ 2
❦ 3
L
❦ 4
β
3
❦ 5 L
y 4
α
x
10 of 12
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
E = 70000N/mm2 , A = 50 mm2 , α = 30 ◦, β = 60 ◦, L = 500 mm, F = 500 N
Task 2 Solution 1. Determine all relevant information for each element e
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Element e
αe
❦ 1 ❦ 2 ❦ 3 ❦ 4 ❦ 5
0◦
−90 −30 −90 −150
◦
◦
◦
◦
sin αe
cos αe
0
1
−1 √ 0 −1/2 3/2 −1 √ 0 −1/2 − 3/2
√ L
e
Ae
E e
3L/2
A
E
L/2
A
E
L
A
E
L L
A A
E E
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
Task 2 Solution 6. Prepare the reduced equation system
− 0 0 F 0 0
12 of 12
=
r ˜
AE 4L
˜a = K˜
√
5 0√ 0 8/ 3 0 0 3 0 −1 0
0 0 8 0 −8
√
3 0 0 6 0
− −1 0 8 0 10
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
a1,y a2,x a2,y a3,x a3,y
Task 2 Solution 6. Prepare the reduced equation system
− 0 0 F 0 0
=
r ˜
AE 4L
√
7. Solve the reduced equation system r˜ =
r
12 of 12
=
433.01 0 0 500 0 0 433.01 500
− −
˜a = K˜
√
−
0 0 8 0 −8
˜ a and K˜
derivation of the vectors
N,
a
=
3 0 0 6 0
−1
5 0√ 0 8/ 3 0 0 3 0 −1 0
0 8 0 10
0 0.0357 0 0.1964 0.0103 0.1607 0 0
− − −
Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of Continuum Mechanics | WS 2015/16
a1,y a2,x a2,y a3,x a3,y
mm
